A new class of nonlinear partial differential equations solvable by quadratures

We introduce a class of nonlinear partial differential equations in two independent scalar variables, say x and t, characterized by the property that the initial value problem for given boundary values can be solved by quadratures. The Liouville equation enjoys such a property and seems to be the most simple equation among the elements of this class. Hence we term these equations generalized Liouville equations. We further introduce the Riccati property, which refers to nonlinear ordinary differential equations and generalizes a well known property of the Riccati equation. This property requires that, whenever one particular solution of an equation is given, then it is possible to construct from that the general solution by quadratures. Nonlinear ordinary differential equations which enjoy the Riccati property are shown to be related to generalized Liouville equations.